Revisiting Gilbert Strang's "A Chaotic Search for $i$"

TL;DR

This paper extends Strang's symbolic analysis of Newton's method for finding complex roots to higher-order methods like Householder and secant, revealing their chaotic dynamics and periodicity through explicit formulas and experiments.

Contribution

It introduces new symbolic formulas for higher-order and secant methods' iteration behavior, expanding Strang's original analysis to a broader class of root-finding algorithms.

Findings

Derived explicit formulas for higher-order and secant methods' iterations.

Illustrated chaotic and periodic behaviors through computational experiments.

Compared complexity of Schr"oder iterations with simpler methods.

Abstract

In the paper "A Chaotic Search for $i$"~(\cite{strang1991chaotic}), Strang completely explained the behaviour of Newton's method when using real initial guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\pm i$. He explored an exact symbolic formula for the iteration, namely $x_{n}=\cot{ \left( 2^{n} \theta_{0} \right) }$, which is valid in exact arithmetic. In this paper, we extend this to to $k^{th}$ order Householder methods, which include Halley's method, and to the secant method. Two formulae, $x_{n}=\cot{ \left( \theta_{n-1}+\theta_{n-2} \right) }$ with $\theta_{n-1}=\mathrm{arccot}{\left(x_{n-1}\right)}$ and $\theta_{n-2}=\mathrm{arccot}{\left(x_{n-2}\right)}$, and $x_{n}=\cot{ \left( (k+1)^{n} \theta_{0} \right) }$ with $\theta_{0} = \mathrm{arccot}(x_{0})$, are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schr\"{o}der iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's \textsl{Fractals[Newton]} package to visualize general one-step iterations by disguising them as Newton iterations.